Communications in Mathematical Sciences
Volume 17 (2019)
Periodic solutions to nonlinear Euler–Bernoulli beam equations
Pages: 2005 – 2034
Bending vibrations of thin beams and plates can be described by nonlinear Euler–Bernoulli beam equation with $x$-dependent coefficients. In this paper we demonstrate the existence of families of time-periodic solutions to such a model by virtue of a Lyapunov–Schmidt reduction together with a Nash–Moser method. This result holds for all parameters $(\epsilon , \omega)$ in a Cantor set with asymptotically full measure as $\epsilon \to 0$.
Euler–Bernoulli beam equations, variable coefficients, periodic solutions, Nash–Moser iteration
2010 Mathematics Subject Classification
35B10, 35L75, 58C15
The research of B.C. was supported in part by NSFC Grant 11901232 and China Postdoctoral Science Foundation Funded Project 2019M651191. The research of Y.G. was supported in part by NSFC grant 11871140, 11671071, JJKH20180006KJ, JLSTDP 20190201154JC and FRFCU2412019BJ005. The research of Y.L. was supported in part by NSFC grant 11571065.
Received 29 April 2018
Accepted 1 July 2019
Published 6 January 2020